Abstract
In this paper, we present hybrid weighted essentially non-oscillatory (WENO) schemes with several discontinuity detectors for solving the compressible ideal magnetohydrodynamics (MHD) equation. Li and Qiu (J Comput Phys 229:8105–8129, 2010) examined effectiveness and efficiency of several different troubled-cell indicators in hybrid WENO methods for Euler gasdynamics. Later, Li et al. (J Sci Comput 51:527–559, 2012) extended the hybrid methods for solving the shallow water equations with four better indicators. Hybrid WENO schemes reduce the computational costs, maintain non-oscillatory properties and keep sharp transitions for problems. The numerical results of hybrid WENO-JS/WENO-M schemes are presented to compare the ability of several troubled-cell indicators with a variety of test problems. The focus of this paper, we propose optimal and reliable indicators for performance comparison of hybrid method using troubled-cell indicators for efficient numerical method of ideal MHD equations. We propose a modified ATV indicator that uses a second derivative. It is advantageous for differential discontinuity detection such as jump discontinuity and kink. A detailed numerical study of one-dimensional and two-dimensional cases is conducted to address efficiency (CPU time reduction and more accurate numerical solution) and non-oscillatory property problems. We demonstrate that the hybrid WENO-M scheme preserves the advantages of WENO-M and the ratio of computational costs of hybrid WENO-M and hybrid WENO-JS is smaller than that of WENO-M and WENO-JS.
Similar content being viewed by others
References
Balsara, D.S.: Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction. Astrophys. J. Suppl. 151, 149–184 (2004)
Balsara, D.S., Spicer, D.S.: A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys. 149, 270–292 (1999)
Balsara, D.S., Shu, C.W.: Monotonicity preserving WENO schemes with increasingly high-order of accuracy. J. Comput. Phys. 160, 405–452 (2000)
Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved WENO scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008)
Brio, M., Wu, C.C.: An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75, 400–422 (1988)
Christlieb, A.J., Rossmanith, J.A., Tang, Q.: Finite difference weighted essentially non-oscillatory schemes with constrained transport for ideal magnetohydrodynamics. J. Comput. Phys. 268, 302–325 (2014)
Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)
Costa, B., Don, W.S.: High order hybrid central-WENO finite difference scheme for conservation laws. J. Comput. Appl. Math. 204, 209–218 (2007)
Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)
Dai, W., Woodward, P.R.: A simple finite difference scheme for multidimensional magnetohydrodynamical equations. J. Comput. Phys. 142, 331–369 (1998)
Fitzpatrick, R.: Plasma Physics: An Introduction. CRC Press, Boca Raton (2015)
Gerolymos, G.A., Scénéhal, D., Vallet, I.: Very-high-order WENO schemes. J. Comput. Phys. 228, 8481–8524 (2009)
Goedbloed, H., Poedts, S.: Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasma. Cambridge University Press, Cambridge (2004)
Ha, Y., Kim, C.H., Lee, Y.J., Yoon, J.: An improved weighted essentially non-oscillatory scheme with a new smoothness indicator. J. Comput. Phys. 232, 68–86 (2013)
Harten, A.: Adaptive multiresolution schemes for shock computations. J. Comput. Phys. 115, 319–338 (1994)
Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high-order accurate non-oscillatory schemes III. J. Comput. Phys. 71, 231–303 (1987)
Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted-essentially-non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)
Hill, D.J., Pullin, D.I.: Hybrid tuned center-difference-WENO method for larged eddy simulations in the presence of strong shocks. J. Comput. Phys. 194, 435–450 (2004)
Jeffrey, A., Taniuti, T.: Non-linear Wave Propagation. Academic Press, New York (1964)
Jiang, G., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Jiang, G., Wu, C.: A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 150, 561–594 (1999)
Krivodonova, L., Xin, J., Remacle, J.-F., Chevaugeon, N., Flaherty, J.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48, 323–338 (2004)
Li, G., Qiu, J.: Hybrid weighted essentially non-oscillatory schemes with different indicators. J. Comput. Phys. 229, 8105–8129 (2010)
Li, G., Qiu, J.: Hybrid WENO Schemes different indicators on curvilinear grids. Adv. Comput. Math. 40, 747–772 (2014)
Li, G., Lu, C., Qiu, J.: Hybrid well-balanced WENO schemes with different indicators for shallow water equations. J. Sci. Comput. 51, 527–559 (2012)
Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)
Orszag, S.A., Tang, C.M.: Small-scale structure of two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 90, 129–143 (1979)
Pirozzoli, S.: Conservative hybrid compact-WENO schemes for shock–turbulence interaction. J. Comput. Phys. 178, 81–117 (2002)
Qiu, J., Shu, C.W.: A comparison of troubled-cell indicator for Runge–Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM J. Sci. Comput. 27, 995–1013 (2005)
Roe, P.L., Balsara, D.S.: Notes on the eigensystem of magnetohydrodynamics. SIAM. J. Appl. Math. 56(1), 57–67 (1996)
Rossmanith, J.: A wave propagation method with constrained transport for shallow water and ideal magnetohydrodynamics, Ph.D Thesis, University of Washington, Seattle (2002)
Rossmanith, J.: An unstaggered high-resolution constrained transport method for magnetohydrodynamic flows SIAM. J. Sci. Comput. 28, 1766–1797 (2006)
Ryu, D., Miniati, F., Jones, T.W., Frank, A.: A divergence-free upwind code for multi-dimensional magnetohydrodynamic flows. Astrophys. J. 509, 244–255 (1998)
Shu, C.W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Quarteroni, A. (ed.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin (1998)
Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)
Tóth, G.: The \(\nabla \cdot B = 0\) constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161, 605–652 (2000)
Zhu, H., Qiu, J.: Adaptive Runge–Kutta discontinuous Galerkin methods using different indicators: one-dimensional case. J. Comput. Phys. 228, 6957–6976 (2009)
Acknowledgements
Youngsoo Ha and Chang Ho Kim were supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2014M1A7A1A03029872), also Youngsoo Ha was supported by the National Research Foundation of Korea (NRF) (NRF-2013R1A1A2013793). Kwang-Il You is supported by Ministry of Science, ICT and Future Planning of Korea (NIST Code: 1711046884). We would like to thank the anonymous reviewers for constructive suggestions to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, C.H., You, KI. & Ha, Y. Hybrid Finite Difference Weighted Essentially Non-oscillatory Schemes for the Compressible Ideal Magnetohydrodynamics Equation. J Sci Comput 74, 607–630 (2018). https://doi.org/10.1007/s10915-017-0462-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-017-0462-3